A simple method for the estimation of thermal inertia

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1 Click Here for Full Article GEOPHYSICAL RESEARCH LETTERS, VOL. 37,, doi: /2009gl041851, 2010 A simle method for the estimation of thermal inertia J. Wang, 1,2 R. L. Bras, 1,2 G. Sivandran, 3 and R. G. Knox 3 Received 19 November 2009; acceted 29 January 2010; ublished 6 March [1] A simle method for estimating thermal inertia using daily amlitude of surface soil heat flux and temerature is roosed. Based on an analytical solution of the diffusion equation where the soil temerature is idealized as a sinusoidal function, the thermal inertia is exressed as a roortional coefficient in a linear equation relating the amlitudes of surface soil (ground) heat flux and temerature. Tests of the roosed method demonstrate that this simle method is otentially useful in the Earth and extra terrestrial remote sensing. Citation: Wang, J., R. L. Bras, G. Sivandran, and R. G. Knox (2010), A simle method for the estimation of thermal inertia, Geohys. Res. Lett., 37,, doi: /2009gl Introduction [2] The thermal inertia, I, is a comosite arameter defined as I ffiffiffiffiffiffiffi ck ; ð1þ 1 Deartment of Civil and Environmental Engineering, University of California, Irvine, California, USA. 2 Formerly at R. M. Parsons Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA. 3 R. M. Parsons Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA. Coyright 2010 by the American Geohysical Union /10/2009GL where k is the thermal conductivity (W m 1 K 1 ), r the bulk density (kg m 3 ), and c the secific heat (J kg 1 K 1 ) of the bulk soil material. I has been identified as the arameter of rimary imortance in the Earth and extraterrestrial remote sensing [Carlson et al., 1981; Palluconi and Kieffer, 1981]. Thermal inertia of lanetary surface materials is the key roerty controlling the diurnal and seasonal surface temerature variations. It reresents a comlex combination of article size, rock abundance, bedrock outcroing and the degree of induration. Direct measurements of these hysical roerties at the global scale is unrealistic for the Earth, and so far imossible for other lanets. Hence in the Earth and extra terrestrial remote sensing, I must be estimated from radiometric observations aided with model simulations. A oular method for the remote sensing of Mars is to fit multile arameters, including I, to brightness temerature observations [Jakosky et al., 2000; Putzig and Mellon, 2007]. [3] The success of the method of Jakosky et al. [2000] and Putzig et al. [2005] for the estimation of I deends critically on modeling of the surface energy balance. Since diurnal variation of surface temerature is directly related to that of surface soil (ground) heat flux [Wang and Bras, 1999], the thermal inertia can be obtained, theoretically, when simultaneous measurements of ground heat flux and surface (skin) temerature are available without knowing other comonents of the surface energy balance such as radiative energy fluxes and turbulent latent and sensible heat fluxes. When radiative fluxes and turbulent heat fluxes are available, the ground heat flux can be derived from the surface energy balance equation. Since the amlitude of the diurnal variation of surface temerature is dominated by the thermal inertia, it is exected that the amlitude alone would rovide essential information about the thermal inertia even though the detailed time variations of the surface temerature is either unknown or unavailable. Sellers [1965] has rovided a simle analytical relationshi linking the thermal inertia arameter to the diurnal/seasonal amlitudes of surface temerature and ground (or surface soil) heat flux records for a sinusoidal function of surface temerature. The goal of this study is to show that Sellers results can be alied to more general conditions, leading to an effective algorithm for the estimation of thermal inertia, in articular using remote sensing observations. 2. Method and Data 2.1. Method [4] Sace time (z t) distribution of soil temerature T(z, t) and soil heat flux Q(z, t) can be obtained by solving a diffusion equation (without heat z 0; ð2þ where z is the vertical coordinate with the origin set at the soil atmoshere interface under certain initial and boundary conditions. Our goal is to derive an analytical exression relating ground heat flux G = Q(0, t) to surface temerature T s = T(0, t) under the least restrictive boundary condition (BC) at the surface and initial condition (IC). In this analysis, no BC and IC are secified at the surface. To ensure an analytical solution, a boundary condition is imosed as Tð 1; tþ ¼ T 0 ; ð3þ where T 0 is a constant. This boundary condition is a statement that surface thermal forcing has a finite enetration deth. For most of soils of the Earth, the enetration deth is no more than 1 m for diurnal cycle and is on the order of 10 m for annual cycle [e.g., Sellers, 1965,. 136]. [5] Although equations (2) (3) would not give a unique solution of T(z, t), they do lead to the desired exression linking Q(z, t) tot(z, t) at the same z. Derivation of the exression is facilitated using a mathematical tool known as fractional calculus [Oldham and Sanier, 1974; Samko et al., 1993; Miller and Ross, 1993; Podlubny, 1999]. Wang and Bras [1999] rovided an examle of using the frac- 1of5

2 [Podlubny, 1999,. 311] by substituting equation (6) into equation (5) that Gt ðþ¼it ffiffiffiffiffi! 0 sin!0 t þ ; ð7þ 4 indicating that G is also a sinusoidal function with a haseshift of /4 relative to T s (i.e. G leads T s by 3 hours for diurnal cycle, and 1.5 months for annual cycle). According to equation (7), the diurnal amlitude of G, DG, relates to that of T s, DT, through G ¼ IT ffiffiffiffiffi! 0: ð8þ Figure 1. (a) Ground heat flux G Q(0, t) (Wm 2 ), and (b) surface (skin) temerature T s T(0, t) ( C)measured at the Lucky Hill site in the Walnut Gulch Watershed, 5 16 June tional calculus to deriving the equations relating G to T s and vice versa. The relationshi between Q(z, t) and T(z, t) is known as the half order (time) z; t Qz; ð tþ ð Þ ¼ Z t sþ ds ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðt sþ where s the dummy integration variable, and I (W m 2 K 1 s 1/2 or J m 2 K 1 s 1/2 called by some the thermal inertia unit or tiu) is given in equation (1). Equation (4), the general solution of equation (2) under boundary condition equation (3), is essentially the same solution derived by Wang and Bras [1999] excet that it does not require a uniform initial vertical rofile of soil temerature. Secifically, equation (4) holds at z = 0 (soil surface), leading to an equation relating ground heat flux G to time history of skin temerature T s (t) =T(0, t), Gt ðþqð0; tþ ¼ I Z t s ðþ ds ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð5þ ðt sþ A numerical algorithm for comuting the singular integral in equation (4) or (5) is given by Bennett et al [2008]. [6] Equation (5) has a simle exression for an idealized function of T s. When T s is sinusoidal in time, i.e., T s ðþ¼t t 0 þ T sin ð! 0 tþ; ð6þ where T 0 is the daily mean surface temerature, DT the diurnal amlitude of surface temerature, and w 0 =2/d with d being the length of day (24 hours). Here T 0 and DT are constant (i.e. indeendent of t). It can be shown [7] Although well known [Sellers, 1965,. 138], equation (7) is derived here as a secial case of equation (4) under less (erhas least) restrictive initial and boundary conditions since the derivation of equation (4) or (5) requires rescribing neither initial soil temerature rofile nor temerature/ flux at the surface, while Sellers analysis is conditioned on a secific initial soil temerature rofile. Nonetheless, our analysis justifies and generalizes the results of Sellers [1965]. Equation (8), a convenient equation from which I can be obtained as the regression coefficient between DG and DT airs (one air er day for examle), is referred to as the simle method. [8] Obviously, the simle method is an aroximation since equation (8) (or equation (7)) is exact only for sinusoidal T s with a constant I. More accurate estimates of I may be obtained using equation (5), which holds for arbitrary T s. We intend to demonstrate the feasibility of the simle method with actual surface soil temerature time series, which are not sinusoidal, from in situ and remote sensing observations. The simle method as an aroximation is also due to the assumtion of constant I. In reality, I varies with soil moisture. It is ossible to derive a more accurate analytical solution of G in terms of T s taking the effect of variable soil moisture into account. Nonetheless, the effect of variable soil moisture on I is relatively weak because of the square root deendence of I on soil moisture. As a result, the simle method, as a first order aroximation, would give an estimate of I corresonding to a reresentative value of soil moisture. To minimize the effect of soil moisture on I, the tests below will use observations made at dry locations. In the analysis below, we consider the estimated I using equation (5) as the truth to which the estimated I using the simle method is comared Data [9] The simle method has been tested using field data. The data of soil heat fluxes and soil temerature used in this study were collected during a field exeriment at the Lucky Hill site ( N, W) in the Walnut Gulch exerimental watershed [Renard et al., 2008] 2 17 June 2008 (Figure 1). The exeriment was during the re monsoon season characterized by strong insolation and occasional clouds. The exerimental site is located at a mostly bare sot over a flat area. The soil does not contain removable moisture from the surface down to at least tens of centimeters after an extensive eriod without rain. Ground heat flux was measured by heat flux lates laced 1 cm below the surface. Skin temerature was measured using an infrared 2of5

3 Table 1. Observed Daily Maxima and Minima of G and T s During 5 15 June 2008 Day 5th 6th 7th 8th 9th 10th 11th 12th 13th 14th 15th I (di) G max (W m 2 ) G min (W m 2 ) T max s ( C) T min s ( C) I est (.041) I obs.827 (.014) See Figure 1. Data for 4 June 2000 are not used (daily minimum heat flux and temerature missing). I est ( 10 3 Wm 2 K 1 s 1/2 ) stands for the estimate using equation (8), where DG = G max G min and DT = T max s T min s, and I obs stand for the estimate as the regression coefficient according to equation (5). I and di stand for mean and standard deviation of the corresonding estimates. thermometer. Soil variables were samled at 10 seconds interval. Ten minute averaged data are used in the following analysis to reduce the samling errors. 3. Validation of the Simle Method [10] The estimated I (W m 2 K 1 s 1/2 tiu) using the simle method, I est, and the observed I, I obs, with their means and standard deviations are given in Table 1. Also listed in Table 1 are the twelve airs of maximum/minimum flux/temerature and twelve samles of estimates of I according to equation (8). I obs, considered the true value of I, for the soil material at the site is obtained as the regression coefficient of equation (5) using the simultaneously measured G and T s shown in Figure 1. The integral on the right hand side of equation (5) is comuted using an algorithm described by Bennett et al [2008]. A standard linear regression rogram is used to comute the mean and standard error where the observed G is the regressed variable and the integral of soil temerature is the error free regressor. The choice of regressor is based on the fact that temerature measurements are more accurate than heat flux measurements. [11] I est has a mean, , slightly larger than that of I obs, I est also has a larger standard error (deviation), , than that of I obs, The larger estimation error of I est is mainly due to the small samle size of 12, comared to the large samle size of 831, i.e. 831 samles of ten minute averaged soil heat flux and soil temerature are used in comuting the regression coefficient according to equation (5). Using the standard t test assuming Gaussian errors, the 95% confidence interval for the difference (or bias), I est I obs, is [0.013, 0.069] 10 3, which excludes zero. Hence the bias, although small, is significant at 5% level. I est is a slight over estimate of the truth with a bias about 5% on average. This bias is likely caused by the fact that the I derived from equation (8) only reresents the effect of the first harmonic comonents of the actual diurnal cycles of G and T s. Nonetheless, the contribution from the higher order harmonic terms, due to rocesses such as clouds and winds, aears to be relatively small considering the samling error of heat flux sensors on the order of at least 10%. For ractical uroses, the simle method gives a good estimate of I. 4. Alications [12] The roosed simle method would be most useful when high frequency samles of temerature and heat fluxes are lacking. It only needs the amlitudes of diurnal cycles of collocated temerature and soil heat flux. The estimates of I using the simle method may be imroved through more accurate information of diurnal variability in temerature and soil heat flux even though the detailed rocesses of temerature and fluxes are oorly known. For examle, the amlitudes of diurnal variations of soil surface heat fluxes may be obtained from the other indeendently measured or modeled comonents in the surface energy budget. Here we resent an examle of estimating I for Martian surfaces where all data are from remote sensing observations. We emhasize that the goal of this analysis is not to rove the sueriority of the simle method. Instead, we merely intend to illustrate its otential as an alternative to the existing ones under certain conditions, hoing that more indeendent tests will follow to justify its usefulness. [13] Estimation of the thermal inertia of Martian surface has been studied by a number of researchers [e.g., Kieffer et al., 1977; Mellon et al., 2000; Putzig and Mellon, 2007]. The basic idea of the existing methods is to fit the modeled temerature to the observations by simultaneously varying several arameters, such as albedo, thermal inertia and dust oacity, that affect the thermal status of Martian surface. The simle method offers a ossibility of estimating I without doing numerical simulations of the detailed heat transort rocesses. The test site of the simle method (50 latitude) is the same one as that of Figure 1 of Mellon et al. [2000]. [14] Since no direct measurements of G on the Martian surface are available, DG will be derived as a residual of the surface energy balance equation, R n ¼ G þ H þ E; with R n, H and E are the net radiation, the turbulent sensible heat flux, and the latent heat flux, resectively. To simlify the matter, we select a mid latitude location where shortwave radiation, emitted long wave radiation and G are the dominant comonents of equation (9). Segal et al [1997] reorted that H over the mid latitude Martian surface is on the order of 20 W m 2, and incoming long wave irradiance, R L d, on the order of 30 W m 2 [Segal et al., 1997]. Since H and R L d are one order of magnitude smaller than the other radiative fluxes, the energy balance as a first aroximation reduced to ð1 ÞR s Ts 4 ð9þ ¼ G; ð10þ where a is the surface albedo, R s the incoming solar irradiance, T s the surface (hysical) temerature, the thermal (infrared) emissivity, and s the Stefan Boltzmann constant. 3of5

4 useful roerty of the simle method for two reasons. First, remote sensing data may not always cature the diurnal maxima/minima of G and T s due to limited visiting frequency of the Mars sacecrafts. Second, time averaged heat flux and temerature have less errors than the original signals. This tye of flexibility makes the simle method the most arameter arsimonious algorithm for estimating I using remote sensing observations of either diurnal or annual amlitudes of surface soil heat flux and temerature. Meanwhile, imroved knowledge of eaks of G and T s may be obtained through further analysis of the existing Mars Global Surveyor data roducts, a toic of future research. Figure 2. Estimated I (circles) using the simle method through equation (8) versus results of Mellon et al [2000] (crosses). DG is comuted using equation (10), where R s = R TOA cos with R TOA = 590 W m 2 the solar radiation at the to of Martian atmoshere (e.g., Aelbaum et al., 1993), and the latitude = 50. In this examle, a = 0.25 and = 1.0 [Mellon et al., 2000]. Then DG can be obtained, according to equation (10), from daily maximum and minimum T s given by the temerature curves in Figure 1 of Mellon et al. [2000]. a = 0.25 is chosen and = 1 based on Table I of Mellon et al. [2000]. [15] Figure 2 shows that the simle method roduces estimates of I in close agreement with the results reorted by Mellon et al. [2000, Figure 1]. The agreement is closer for relatively large diurnal amlitude of T s, e.g. DT >90K.We notice that the new estimates of I using the simle method are smaller than those obtained by Mellon et al. [2000]. This is consistent with the aroximation of neglecting R L d, H and E in equation (9) or equation (2) of Mellon et al. [2000], which leads to underestimates of DG according to equation (10). Meanwhile, the simle method under estimates I no more than 15% relative to those reorted by Mellon et al. [2000, Figure 1] suggests that the simle method would roduce better estimates of I when the data of R L d, H and E are available. We emhasize that the comarison does not necessarily quantify the erformance of the simle method. It does show that equation (8) is a convenient algorithm for a first order estimation of I, which may be imroved with more accurate inut of DG. A major advantage of the simle method is that it does not require inut of the local gradient of soil temerature, which is difficult to measure even in field exeriment and certainly not available from remote sensing observation. The above analysis indicates that the simle method is able to estimate I using remote sensing observations only without direct measurement of G. We exect the simle method could roduce even better estimates of I if direct measurements of G using flux lates can be made in the future Mars missions. [16] Theoretically, the simle method can also estimate I using annual cycles of daily mean records of DG and DT since the time averaged soil temerature is described by the same diffusion equations (with constant arameters), i.e., equation (2) [Bennett et al., 2008]. This is an esecially 5. Conclusions [17] The roosed simle method yields good estimates of thermal inertia. The major advantage of the simle method lies in the minimum inut data available from remote sensing observations. The simle method is a romising alternative algorithm suitable for deriving global mas of thermal inertia for both the Earth and the other lanets, articularly when comlete diurnal and annual records of temerature and surface soil heat fluxes are lacking. [18] Acknowledgments. This study was suorted by the ARO grant W911NF , the NASA grants NNX07AD29D and NNG05GA17G, and the NSF grant ATM The field exeriment equiment was rovided by the ARO roject W911NF We thank Dave Goodrich and John Smith of USDA ARS for their suort during the field exeriment. References Alebaum, J., G. A. Landis, and I. Sherman (1993), Solar radiation on Mars Udate 1991, Sol. Energy, 50(1), Bennett,W.B.,J.Wang,andR.L.Bras (2008), Estimation of global ground heat flux, J. Hydrometeorol., 9(4), Carlson, T. Y., J. K. Dodd, S. G. Benjamin, and J. N. Cooer (1981), Satellite estimation of the surface energy balance, moisture availability and thermal inertia, J. Al. Meteorol., 20, Jakosky, B. M., M. T. Mellon, H. H. Kieffer, P. R. Christensen, E. S. Varnes, and S. W. Lee (2000), The thermal inertia of Mars from the Mars global surveyor thermal emission sectrometer, J. Geohys. Res., 105, Kieffer, H. H., T. Z. Martin, A. R. Peterfreund, B. M. Jakosky, E. D. Miner, and F. D. 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